Graph Search Methods Graph Search Methods A search method starts - - PDF document

SLIDE 1

Graph Search Methods

• A vertex u is reachable from vertex v iff there is a

path from v to u.

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Graph Search Methods

• A search method starts at a given vertex v and

visits/labels/marks every vertex that is reachable from v.

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Graph Search Methods

• Many graph problems solved using a search

method.

Path from one vertex to another. Is the graph connected? Find a spanning tree. Etc.

• Commonly used search methods:

Breadth-first search. Depth-first search.

Breadth-First Search

• Visit start vertex and put into a FIFO queue.
• Repeatedly remove a vertex from the queue, visit

its unvisited adjacent vertices, put newly visited vertices into the queue.

Breadth-First Search Example

Start search at vertex 1.

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Breadth-First Search Example

Visit/mark/label start vertex and put in a FIFO queue.

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FIFO Queue 1

SLIDE 2

Breadth-First Search Example

Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue 1

Breadth-First Search Example

Remove 1 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2

2

4

4

Breadth-First Search Example

Remove 2 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2

2

4

4

Breadth-First Search Example

Remove 2 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4

4

5

5

3

3

6

6

Breadth-First Search Example

Remove 4 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4

4

5

5

3

3

6

6

Breadth-First Search Example

Remove 4 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5

5

3

3

6

6

SLIDE 3

Breadth-First Search Example

Remove 5 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5

5

3

3

6

6

Breadth-First Search Example

Remove 5 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3

3

6

6

9

9

7

7

Breadth-First Search Example

Remove 3 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3

3

6

6

9

9

7

7

Breadth-First Search Example

Remove 3 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6

6

9

9

7

7

Breadth-First Search Example

Remove 6 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6

6

9

9

7

7

Breadth-First Search Example

Remove 6 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6 9

9

7

7

SLIDE 4

Breadth-First Search Example

Remove 9 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6 9

9

7

7

Breadth-First Search Example

Remove 9 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6 9 7

7

8

8

Breadth-First Search Example

Remove 7 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6 9 7

7

8

8

Breadth-First Search Example

Remove 7 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6 9 7 8

8

Breadth-First Search Example

Remove 8 from Q; visit adjacent unvisited vertices; put in Q.

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FIFO Queue

2 4 5 3 6 9 7 8

8

Breadth-First Search Example

Queue is empty. Search terminates.

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FIFO Queue

2 4 5 3 6 9 7 8

SLIDE 5

Breadth-First Search Property

• All vertices reachable from the start vertex

(including the start vertex) are visited.

Time Complexity

• Each visited vertex is put on (and so

removed from) the queue exactly once.

• When a vertex is removed from the queue,

we examine its adjacent vertices.

O(n) if adjacency matrix used O(vertex degree) if adjacency lists used

• Total time

O(mn), where m is number of vertices in the component that is searched (adjacency matrix)

Time Complexity

O(n + sum of component vertex degrees) (adj. lists) = O(n + number of edges in component)

Path From Vertex v To Vertex u

• Start a breadth-first search at vertex v.
• Terminate when vertex u is visited or when

Q becomes empty (whichever occurs first).

• Time

O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number

• f edges)

Is The Graph Connected?

• Start a breadth-first search at any vertex of

the graph.

• Graph is connected iff all n vertices get

visited.

• Time

O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number

• f edges)

Connected Components

• Start a breadth-first search at any as yet

unvisited vertex of the graph.

• Newly visited vertices (plus edges between

them) define a component.

• Repeat until all vertices are visited.

SLIDE 6

Connected Components

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Time Complexity

O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number of edges)

Spanning Tree

Breadth-first search from vertex 1.

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Breadth-first spanning tree.

Spanning Tree

• Start a breadth-first search at any vertex of

the graph.

• If graph is connected, the n-1 edges used to

get to unvisited vertices define a spanning tree (breadth-first spanning tree).

• Time

O(n2) when adjacency matrix used O(n+e) when adjacency lists used (e is number

• f edges)

Depth-First Search

depthFirstSearch(v) { Label vertex v as reached. for (each unreached vertex u adjacenct from v) depthFirstSearch(u); }

Depth-First Search Example

Start search at vertex 1.

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Label vertex 1 and do a depth first search from either 2 or 4.

1 2

Suppose that vertex 2 is selected.

SLIDE 7

Depth-First Search Example

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Label vertex 2 and do a depth first search from either 3, 5, or 6.

1 2 5

Suppose that vertex 5 is selected.

Depth-First Search Example

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Label vertex 5 and do a depth first search from either 3, 7, or 9.

1 2 5 9

Suppose that vertex 9 is selected.

Depth-First Search Example

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Label vertex 9 and do a depth first search from either 6 or 8.

1 2 5 9 8

Suppose that vertex 8 is selected.

Depth-First Search Example

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Label vertex 8 and return to vertex 9.

1 2 5 9 8

From vertex 9 do a dfs(6).

6

Depth-First Search Example

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Label vertex 6 and do a depth first search from either 4 or 7.

6 4

Suppose that vertex 4 is selected.

Depth-First Search Example

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Label vertex 4 and return to 6.

6 4

From vertex 6 do a dfs(7).

7

SLIDE 8

Depth-First Search Example

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Label vertex 7 and return to 6.

6 4 7

Return to 9.

Depth-First Search Example

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Return to 5.

Depth-First Search Example

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Do a dfs(3).

3

Depth-First Search Example

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Label 3 and return to 5.

3

Return to 2.

Depth-First Search Example

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Return to 1.

Depth-First Search Example

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Return to invoking method.

SLIDE 9

Depth-First Search Properties

• Same complexity as BFS.
• Same properties with respect to path

finding, connected components, and spanning trees.

• Edges used to reach unlabeled vertices

define a depth-first spanning tree when the graph is connected.

• There are problems for which bfs is better

than dfs and vice versa.